Integrand size = 18, antiderivative size = 124 \[ \int \frac {\cosh \left (a+b \sqrt {c+d x}\right )}{x} \, dx=\cosh \left (a+b \sqrt {c}\right ) \text {Chi}\left (b \left (\sqrt {c}-\sqrt {c+d x}\right )\right )+\cosh \left (a-b \sqrt {c}\right ) \text {Chi}\left (b \left (\sqrt {c}+\sqrt {c+d x}\right )\right )-\sinh \left (a+b \sqrt {c}\right ) \text {Shi}\left (b \left (\sqrt {c}-\sqrt {c+d x}\right )\right )+\sinh \left (a-b \sqrt {c}\right ) \text {Shi}\left (b \left (\sqrt {c}+\sqrt {c+d x}\right )\right ) \]
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Time = 0.22 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {5473, 5401, 3384, 3379, 3382} \[ \int \frac {\cosh \left (a+b \sqrt {c+d x}\right )}{x} \, dx=\cosh \left (a+b \sqrt {c}\right ) \text {Chi}\left (b \left (\sqrt {c}-\sqrt {c+d x}\right )\right )+\cosh \left (a-b \sqrt {c}\right ) \text {Chi}\left (b \left (\sqrt {c}+\sqrt {c+d x}\right )\right )-\sinh \left (a+b \sqrt {c}\right ) \text {Shi}\left (b \left (\sqrt {c}-\sqrt {c+d x}\right )\right )+\sinh \left (a-b \sqrt {c}\right ) \text {Shi}\left (b \left (\sqrt {c}+\sqrt {c+d x}\right )\right ) \]
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Rule 3379
Rule 3382
Rule 3384
Rule 5401
Rule 5473
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {\cosh \left (a+b \sqrt {x}\right )}{-c+x} \, dx,x,c+d x\right ) \\ & = 2 \text {Subst}\left (\int \frac {x \cosh (a+b x)}{-c+x^2} \, dx,x,\sqrt {c+d x}\right ) \\ & = 2 \text {Subst}\left (\int \left (-\frac {\cosh (a+b x)}{2 \left (\sqrt {c}-x\right )}+\frac {\cosh (a+b x)}{2 \left (\sqrt {c}+x\right )}\right ) \, dx,x,\sqrt {c+d x}\right ) \\ & = -\text {Subst}\left (\int \frac {\cosh (a+b x)}{\sqrt {c}-x} \, dx,x,\sqrt {c+d x}\right )+\text {Subst}\left (\int \frac {\cosh (a+b x)}{\sqrt {c}+x} \, dx,x,\sqrt {c+d x}\right ) \\ & = \cosh \left (a-b \sqrt {c}\right ) \text {Subst}\left (\int \frac {\cosh \left (b \sqrt {c}+b x\right )}{\sqrt {c}+x} \, dx,x,\sqrt {c+d x}\right )-\cosh \left (a+b \sqrt {c}\right ) \text {Subst}\left (\int \frac {\cosh \left (b \sqrt {c}-b x\right )}{\sqrt {c}-x} \, dx,x,\sqrt {c+d x}\right )+\sinh \left (a-b \sqrt {c}\right ) \text {Subst}\left (\int \frac {\sinh \left (b \sqrt {c}+b x\right )}{\sqrt {c}+x} \, dx,x,\sqrt {c+d x}\right )+\sinh \left (a+b \sqrt {c}\right ) \text {Subst}\left (\int \frac {\sinh \left (b \sqrt {c}-b x\right )}{\sqrt {c}-x} \, dx,x,\sqrt {c+d x}\right ) \\ & = \cosh \left (a-b \sqrt {c}\right ) \text {Chi}\left (b \left (\sqrt {c}+\sqrt {c+d x}\right )\right )+\cosh \left (a+b \sqrt {c}\right ) \text {Chi}\left (b \sqrt {c}-b \sqrt {c+d x}\right )+\sinh \left (a-b \sqrt {c}\right ) \text {Shi}\left (b \left (\sqrt {c}+\sqrt {c+d x}\right )\right )-\sinh \left (a+b \sqrt {c}\right ) \text {Shi}\left (b \sqrt {c}-b \sqrt {c+d x}\right ) \\ \end{align*}
Time = 0.29 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.02 \[ \int \frac {\cosh \left (a+b \sqrt {c+d x}\right )}{x} \, dx=\frac {1}{2} e^{-a-b \sqrt {c}} \left (\operatorname {ExpIntegralEi}\left (b \left (\sqrt {c}-\sqrt {c+d x}\right )\right )+e^{2 \left (a+b \sqrt {c}\right )} \operatorname {ExpIntegralEi}\left (b \left (-\sqrt {c}+\sqrt {c+d x}\right )\right )+e^{2 b \sqrt {c}} \operatorname {ExpIntegralEi}\left (-b \left (\sqrt {c}+\sqrt {c+d x}\right )\right )+e^{2 a} \operatorname {ExpIntegralEi}\left (b \left (\sqrt {c}+\sqrt {c+d x}\right )\right )\right ) \]
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\[\int \frac {\cosh \left (a +b \sqrt {d x +c}\right )}{x}d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 217 vs. \(2 (102) = 204\).
Time = 0.27 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.75 \[ \int \frac {\cosh \left (a+b \sqrt {c+d x}\right )}{x} \, dx=\frac {1}{2} \, {\left ({\rm Ei}\left (\sqrt {d x + c} b - \sqrt {b^{2} c}\right ) + {\rm Ei}\left (-\sqrt {d x + c} b + \sqrt {b^{2} c}\right )\right )} \cosh \left (a + \sqrt {b^{2} c}\right ) + \frac {1}{2} \, {\left ({\rm Ei}\left (\sqrt {d x + c} b + \sqrt {b^{2} c}\right ) + {\rm Ei}\left (-\sqrt {d x + c} b - \sqrt {b^{2} c}\right )\right )} \cosh \left (-a + \sqrt {b^{2} c}\right ) + \frac {1}{2} \, {\left ({\rm Ei}\left (\sqrt {d x + c} b - \sqrt {b^{2} c}\right ) - {\rm Ei}\left (-\sqrt {d x + c} b + \sqrt {b^{2} c}\right )\right )} \sinh \left (a + \sqrt {b^{2} c}\right ) - \frac {1}{2} \, {\left ({\rm Ei}\left (\sqrt {d x + c} b + \sqrt {b^{2} c}\right ) - {\rm Ei}\left (-\sqrt {d x + c} b - \sqrt {b^{2} c}\right )\right )} \sinh \left (-a + \sqrt {b^{2} c}\right ) \]
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\[ \int \frac {\cosh \left (a+b \sqrt {c+d x}\right )}{x} \, dx=\int \frac {\cosh {\left (a + b \sqrt {c + d x} \right )}}{x}\, dx \]
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\[ \int \frac {\cosh \left (a+b \sqrt {c+d x}\right )}{x} \, dx=\int { \frac {\cosh \left (\sqrt {d x + c} b + a\right )}{x} \,d x } \]
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\[ \int \frac {\cosh \left (a+b \sqrt {c+d x}\right )}{x} \, dx=\int { \frac {\cosh \left (\sqrt {d x + c} b + a\right )}{x} \,d x } \]
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Timed out. \[ \int \frac {\cosh \left (a+b \sqrt {c+d x}\right )}{x} \, dx=\int \frac {\mathrm {cosh}\left (a+b\,\sqrt {c+d\,x}\right )}{x} \,d x \]
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